Nuprl Lemma : strong-continuity-test-unroll
∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕ?)]. ∀[n:ℕ]. ∀[f,b:Top].
(strong-continuity-test(M;n;f;b) ~ if (n =z 0) then b
if isl(M (n - 1) f) then inr Ax
else strong-continuity-test(M;n - 1;f;b)
fi )
Proof
Definitions occuring in Statement :
strong-continuity-test: strong-continuity-test(M;n;f;b),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
isl: isl(x),
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inr: inr x ,
union: left + right,
subtract: n - m,
natural_number: $n,
universe: Type,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
strong-continuity-test: strong-continuity-test(M;n;f;b),
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
top_wf,
nat_wf,
int_seg_wf,
unit_wf2,
primrec-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eq_int_wf,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
functionEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
unionEquality,
universeEquality,
isect_memberFormation,
axiomSqEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (\mBbbN{}?)]. \mforall{}[n:\mBbbN{}]. \mforall{}[f,b:Top].
(strong-continuity-test(M;n;f;b) \msim{} if (n =\msubz{} 0) then b
if isl(M (n - 1) f) then inr Ax
else strong-continuity-test(M;n - 1;f;b)
fi )
Date html generated:
2019_06_20-PM-02_50_01
Last ObjectModification:
2018_09_26-AM-09_54_16
Theory : continuity
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